First of all, I this that this was my fault. I reversed my family members first and last name on the booking for some reason.
I am sure that the first names and family names in shifts, because Finnair's page I can log only a first name basis.

I have contacted with Finnair and they told me i had to pay 25 euros for a name change (which I would be happy to pay). BUT Finnair also said, they or I can't change anything, because tickets are booked through an online travelagency and only agent can request a changes.

But as I wrote in the first post, the agent wants total over 4,000 euro. which sounds ridiculous to me!